Is there any three-dimensional graph, such that at least two edges must always intersect? Here I may have countably many vertices.
I feel there should be a similar version of $K_{3,3}$ or $K_5$ for the three dimensional case, but I can't quite picture it. I guess in the finite case it suffices to argue that the edges of $K_{3,3}$ or $K_5$ are 'easily unintersected' in the 3D case. What happens in the countable case? For example, when the vertices of a complete graph are composed of all points of the type $(x,y,z)$, $\forall x,y,z\in\mathbb{Z}$?
By three dimensional I only mean that the set of vertices (points) and edges (lines) is a subset of, for example, $\mathbb{R}^3$. I'm not concerned about simplices. Also, I'm naturally not interested in cases where an isomorphic graph does not intersect.
Obviously not a finite graph, or even a graph with countably many vertices.